Satellites and orbits

A satellite is a projectile that never lands. This page shows how that's possible and derives the speed and period of any circular orbit, then applies them to the two orbits the syllabus names: low Earth orbit and geostationary orbit.

Newton's cannon

Newton imagined a cannon on a mountain so tall it pokes out of the atmosphere, firing horizontally. A slow shot falls to the ground nearby. A faster shot lands further away, and because Earth is curved, the ground falls away beneath it as it travels. At just the right speed, the shot falls towards Earth at exactly the rate the surface curves away. It falls forever without getting closer: it is in orbit. Faster still, the orbit becomes an ellipse, and above escape speed the shot leaves Earth for good.

So orbiting is falling. A satellite's acceleration is $g$ at its height, directed at Earth's centre. That is exactly the centripetal acceleration it needs to follow its circle.

Orbital speed and period

For a satellite of mass $m$ in a circular orbit of radius $r$ around a planet of mass $M$, gravity provides the centripetal force:

$\dfrac{GMm}{r^2} = \dfrac{mv^2}{r} \quad\Rightarrow\quad v = \sqrt{\dfrac{GM}{r}}$

Combining with $v = 2\pi r/T$ gives the period:

$T = 2\pi\sqrt{\dfrac{r^3}{GM}}$

The satellite's mass cancels. A 1 kg CubeSat and the 420-tonne ISS at the same height travel at the same speed. Higher orbits are slower and have longer periods. Remember that $r$ is measured from the centre of the planet.

Low Earth orbit and geostationary orbit

A geostationary satellite has a period equal to Earth's rotation, orbits above the equator, and moves in the same direction Earth spins. It stays over the same point on the ground. Because $T$ fixes $r$, there is only one possible geostationary altitude, and that ring of space is crowded with satellites.

The accepted value is 35 786 km. The small difference comes from rounding $M_E$ and from using 24 h. Earth actually turns once relative to the stars every 23 h 56 min (a sidereal day); the extra four minutes in a solar day are because Earth has also moved along its orbit around the Sun.

Apparent weightlessness

Astronauts on the ISS float, but gravity there is still 88% of its surface value. What they've lost is apparent weight, the push of the floor on their feet, which is what bathroom scales measure and what your body senses as weight. The astronaut and the station fall towards Earth with the same acceleration, so the floor never needs to push. The same thing happens briefly at the top of a roller-coaster hill, in a falling lift, or in the "vomit comet" aircraft that flies parabolas to train astronauts.

Orbital decay

In low Earth orbit there is still a trace of atmosphere. Drag removes energy, and the satellite drops gradually into a lower orbit. (Strangely, it speeds up as it does, because lower orbits are faster.) The process accelerates as the air gets thicker, and eventually the satellite re-enters and mostly burns up. The ISS loses about 2 km of height a month and is regularly boosted by engines on docked spacecraft.

When the Sun is active, it heats and expands the upper atmosphere, increasing drag. That is what brought down Skylab, the first US space station, sooner than expected. In July 1979 it broke up over the Indian Ocean and scattered pieces across the area around Esperance, in Western Australia. The local shire issued NASA a tongue-in-cheek 400-dollar fine for littering.